# American Institute of Mathematical Sciences

February  2019, 39(2): 1185-1203. doi: 10.3934/dcds.2019051

## Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity

 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: Hua Chen

Received  June 2018 Revised  August 2018 Published  November 2018

Fund Project: This work is supported by the NSFC under the grant 11631011.

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity $u_t-\triangle_{X} u_t-\triangle_{X} u=u\log|u|$, where $X=(X_1, X_2, ··· , X_m)$ is an infinitely degenerate system of vector fields, and $\triangle_{X}:=\sum^{m}_{j=1}X^{2}_{j}$ is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique, the logarithmic Sobolev inequality and Poincaré inequality, we obtain the global existence and blow-up at $+∞$ of solutions with low initial energy or critical initial energy, and discuss the asymptotic behavior of the solutions.

Citation: Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
##### References:
 [1] J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473. [2] G. Barenblat, I. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303. [3] E. D. Benedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854.  doi: 10.1512/iumj.1981.30.30062. [4] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032. [5] H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations, 24 (1977), 412-425.  doi: 10.1016/0022-0396(77)90009-2. [6] Y. Cao, J. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246 (2009), 4568-4590.  doi: 10.1016/j.jde.2009.03.021. [7] H. Chen and K. Li, The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equations, Math Nach, 282 (2009), 368-385.  doi: 10.1002/mana.200710742. [8] H. Chen, P. Luo and G. Liu, Global solution and blow up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030. [9] H. Chen, P. Luo and S. Tian, Existence and regularity of multiple solutions forinfinitely degenerate nonlinear elliptic equations with singular potential, J. Differ Equ., 257 (2014), 3300-3333.  doi: 10.1016/j.jde.2014.06.014. [10] H. Chen, P. Luo and S. Tian, Multiplicity and regularity of solutions for infinitely degenerate elliptic equations with a free perturbation, J. Math. Pures Appl., 103 (2015), 849-867.  doi: 10.1016/j.matpur.2014.09.004. [11] H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 285 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038. [12] M. Christ, Hypoellipticity in the infinitely degenerate regime, Complex Analysis and Geometry (Columbus, OH, 1999), 59-84, Ohio State Univ. Math. Res. Inst. Publ., 9, de Gruyter, Berlin, 2001. [13] C. David and W. Jet, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl., 69 (1979), 411-418.  doi: 10.1016/0022-247X(79)90152-5. [14] V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49 (1972/73), 57-78.  doi: 10.1007/BF00281474. [15] L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081. [16] S. Ji, J. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 261 (2016), 5446-5464.  doi: 10.1016/j.jde.2016.08.017. [17] J. J. Kohn, Hypoellipticity of some degenerate subelliptic operators, J. Funct. Anal., 159 (1998), 203-216.  doi: 10.1006/jfan.1998.3289. [18] J. J. Kohn, Hypoellipticity at points of infinite type, Proceedings of the International Conference on Analysis, Geometry, Number Theory in honor of Leon Ehrenpreis (Philadelphia, 1998), Contemp. Math., 251 (2000), 393-398.  doi: 10.1090/conm/251/03882. [19] M. Koike, A note on hypoellipticity for degenerate elliptic operators, Publ. Res. Inst. Math. Sci., 27 (1991), 995-1000.  doi: 10.2977/prims/1195169008. [20] V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Mason-John Wiley, Paris, 1994. [21] M. O. Korpusov and A. G. Sveshnikov, Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics, Zh. Vychisl. Mat. Mat. Fiz., 43 (2003), 1835-1869 (in Russian); transl. in: Comput. Math. Math. Phys., 43 (2003), 1765-1797. [22] M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of Sobolev-type nonlinear equations with cubic sources, Differ. Uravn., 42 (2006), 404-415 (in Russian); transl. in: Differ. Equ., 42 (2006), 431-443. doi: 10.1134/S001226610603013X. [23] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt} =-Au+\mathcal{F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814. [24] Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011. [25] Y. Morimoto, On the hypoellipticity for infinitely degenerate semi-elliptic operators, J. Math. Soc. Jpn., 30 (1978), 327-358.  doi: 10.2969/jmsj/03020327. [26] Y. Morimoto, A criterion for hypoellipticity of second order differential operators, Osaka J. Math., 24 (1987), 651-675. [27] Y. Morimoto, Hypoellipticity for infinitely degenerate elliptic operators, Osaka J. Math., 24 (1987), 13-35. [28] Y. Morimoto and T. Morioka, The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators, Bull. Sci. Math., 121 (1997), 507-547. [29] Y. Morimoto and T. Morioka, Hypoellipticity for elliptic operators with infinite degeneracy, Partial Differential Equations and Their Applications (H. Chen and L. Rodino, eds.), World Sci. Publishing, River Edge, NJ, (1999), 240-259. [30] Y. Morimoto and C. Xu, Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators, Astérisque, 284 (2003), 245-264. [31] V. Padrón, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3. [32] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595. [33] M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities, Nonlinear Anal., 66 (2007), 2653-2675.  doi: 10.1016/j.na.2006.03.046. [34] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942. [35] R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.  doi: 10.1137/0501001. [36] S. L. Sobolev, On a new problem of mathematical physics, Izv. Akad. Nauk SSSR Ser. Math., 18 (1954), 3-50. [37] T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal., 14 (1963), 1-26.  doi: 10.1007/BF00250690. [38] T. W. Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21 (1969), 440-453.  doi: 10.2969/jmsj/02130440. [39] R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.

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##### References:
 [1] J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser., 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473. [2] G. Barenblat, I. Zheltov and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech., 24 (1960), 1286-1303. [3] E. D. Benedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854.  doi: 10.1512/iumj.1981.30.30062. [4] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032. [5] H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations, 24 (1977), 412-425.  doi: 10.1016/0022-0396(77)90009-2. [6] Y. Cao, J. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246 (2009), 4568-4590.  doi: 10.1016/j.jde.2009.03.021. [7] H. Chen and K. Li, The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equations, Math Nach, 282 (2009), 368-385.  doi: 10.1002/mana.200710742. [8] H. Chen, P. Luo and G. Liu, Global solution and blow up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030. [9] H. Chen, P. Luo and S. Tian, Existence and regularity of multiple solutions forinfinitely degenerate nonlinear elliptic equations with singular potential, J. Differ Equ., 257 (2014), 3300-3333.  doi: 10.1016/j.jde.2014.06.014. [10] H. Chen, P. Luo and S. Tian, Multiplicity and regularity of solutions for infinitely degenerate elliptic equations with a free perturbation, J. Math. Pures Appl., 103 (2015), 849-867.  doi: 10.1016/j.matpur.2014.09.004. [11] H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 285 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038. [12] M. Christ, Hypoellipticity in the infinitely degenerate regime, Complex Analysis and Geometry (Columbus, OH, 1999), 59-84, Ohio State Univ. Math. Res. Inst. Publ., 9, de Gruyter, Berlin, 2001. [13] C. David and W. Jet, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl., 69 (1979), 411-418.  doi: 10.1016/0022-247X(79)90152-5. [14] V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal., 49 (1972/73), 57-78.  doi: 10.1007/BF00281474. [15] L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081. [16] S. Ji, J. Yin and Y. Cao, Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 261 (2016), 5446-5464.  doi: 10.1016/j.jde.2016.08.017. [17] J. J. Kohn, Hypoellipticity of some degenerate subelliptic operators, J. Funct. Anal., 159 (1998), 203-216.  doi: 10.1006/jfan.1998.3289. [18] J. J. Kohn, Hypoellipticity at points of infinite type, Proceedings of the International Conference on Analysis, Geometry, Number Theory in honor of Leon Ehrenpreis (Philadelphia, 1998), Contemp. Math., 251 (2000), 393-398.  doi: 10.1090/conm/251/03882. [19] M. Koike, A note on hypoellipticity for degenerate elliptic operators, Publ. Res. Inst. Math. Sci., 27 (1991), 995-1000.  doi: 10.2977/prims/1195169008. [20] V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Mason-John Wiley, Paris, 1994. [21] M. O. Korpusov and A. G. Sveshnikov, Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics, Zh. Vychisl. Mat. Mat. Fiz., 43 (2003), 1835-1869 (in Russian); transl. in: Comput. Math. Math. Phys., 43 (2003), 1765-1797. [22] M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of Sobolev-type nonlinear equations with cubic sources, Differ. Uravn., 42 (2006), 404-415 (in Russian); transl. in: Differ. Equ., 42 (2006), 431-443. doi: 10.1134/S001226610603013X. [23] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt} =-Au+\mathcal{F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814. [24] Y. Liu and J. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64 (2006), 2665-2687.  doi: 10.1016/j.na.2005.09.011. [25] Y. Morimoto, On the hypoellipticity for infinitely degenerate semi-elliptic operators, J. Math. Soc. Jpn., 30 (1978), 327-358.  doi: 10.2969/jmsj/03020327. [26] Y. Morimoto, A criterion for hypoellipticity of second order differential operators, Osaka J. Math., 24 (1987), 651-675. [27] Y. Morimoto, Hypoellipticity for infinitely degenerate elliptic operators, Osaka J. Math., 24 (1987), 13-35. [28] Y. Morimoto and T. Morioka, The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators, Bull. Sci. Math., 121 (1997), 507-547. [29] Y. Morimoto and T. Morioka, Hypoellipticity for elliptic operators with infinite degeneracy, Partial Differential Equations and Their Applications (H. Chen and L. Rodino, eds.), World Sci. Publishing, River Edge, NJ, (1999), 240-259. [30] Y. Morimoto and C. Xu, Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators, Astérisque, 284 (2003), 245-264. [31] V. Padrón, Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756.  doi: 10.1090/S0002-9947-03-03340-3. [32] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595. [33] M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities, Nonlinear Anal., 66 (2007), 2653-2675.  doi: 10.1016/j.na.2006.03.046. [34] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942. [35] R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.  doi: 10.1137/0501001. [36] S. L. Sobolev, On a new problem of mathematical physics, Izv. Akad. Nauk SSSR Ser. Math., 18 (1954), 3-50. [37] T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Ration. Mech. Anal., 14 (1963), 1-26.  doi: 10.1007/BF00250690. [38] T. W. Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21 (1969), 440-453.  doi: 10.2969/jmsj/02130440. [39] R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.
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